First, recall that \(\alpha ^{\rm {BB1}}\le \alpha ^{\rm {BB2}}\)  and therefore \(\beta ^{\rm {BB2}}\le \beta ^{\rm {BB1}}\) ; in fact (see, e.g., [1]})
\( \alpha ^{\rm {BB1}}/ \alpha ^{\rm {BB2}}= \beta ^{\rm {BB2}}/ \beta ^{\rm {BB1}}= \cos ^2(\mathbf {s}_{k-1}, A\mathbf {s}_{k-1}).\) 
